The representability problem for the affine-plane Steiner system on nine vertices

Let $X_{2,9}$ be the simplicial complex whose missing faces form a Steiner $(2,3,9)$-system, namely the lines of the affine plane of order $3$.

Affine-plane representability problem. Determine whether

$$\operatorname{rep}(X_{2,9})\leq 5.$$

The source presents this as a problem whose solution could provide a modest step toward the representability bound conjecture. No resolution is given in the supplied text.